Linear Spaces
Convergence
Definition: Let (V,F) be a vector space. A sequence of vectors {xn​}n=1∞​ in V is said to converge to a vector x∈V if for every ϵ>0 there exists an integer N such that ∥xn​−x∥<ϵ for all n≥N. In this case we write xn​→x as n→∞.
Remark: The sequence is said to be convergent if it converges to some vector x∈V. Otherwise, it is said to be divergent.
Example: Let V=R  ∥v∥=v, consider the sequence
{(21​)n}n=1∞​
Proove that this sequence converges to 0 as n→∞.
 Proof: DO IT LATER
Definition: Let (V,F,∥.∥) be a normed space. A sequence of vectors {xn​}n=1∞​ in V is said to be a Cauchy sequence if for every ϵ>0 there exists an integer N such that ∥xn​−xm​∥<ϵ for all n,m≥N.
- Proof: ∥xn​−xm​∥<ϵ for all n,m≥N implies ∥xn​−x∥<ϵ for all n≥N.
#EE501 - Linear Systems Theory at METU